Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

TL;DR

The paper addresses locating a $(1-\varepsilon)$-approximately maximum weight matching in bipartite graphs. It introduces a multiplicative auction algorithm that updates dual prices multiplicatively, achieving $O(m\varepsilon^{-1})$ time and avoiding weight-scale dependence typical of prior methods. The approach yields a simple primal-dual method and extends to a dynamic setting where one side may have vertex deletions and the other side vertex insertions with edges provided in sorted order, maintaining the approximation in $O(m\varepsilon^{-1})$ total time. This offers a faster, more practical alternative to existing algorithms and broadens applicability to dynamic and streaming contexts.

Abstract

$\newcommand{\eps}{\varepsilon}$We present an auction algorithm using multiplicative instead of constant weight updates to compute a $(1-\eps)$-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges in time $O(m\eps^{-1})$, beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM '14] that runs in $O(m\eps^{-1}\log \eps^{-1})$. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a $(1-\eps)$-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time used is $O(m\eps^{-1})$, where $m$ is the sum of the number of initially existing and inserted edges.

Figures (1)

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Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
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• Lemma 2.1: Lemma 2.3 of DuanP14
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• Lemma 2.2